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MTH 102 MTH 102 Past Questions Objective Question
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Question

Evaluate ∫ cos3 x dx
Options
A sin <i>x</i> - (1/3) sin<sup>3</sup> <i>x</i> + C
Correct Answer
B cos <i>x</i> + (1/3) cos<sup>3</sup> <i>x</i> + C
C sec <i>x</i> - (1/3) sec<sup>3</sup> <i>x</i> + C
D tan <i>x</i> - (1/3) tan<sup>3</sup> <i>x</i> + C
Correct Answer

Option A is the correct answer.

Detailed Explanation

To evaluate ∫ cos3 x dx, we use the trigonometric identity cos2x = 1 - sin2x.∫ cos3 x dx = ∫ cos2 x * cos x dx= ∫ (1 - sin2 x) cos x dxNow, perform a substitution. Let u = sin x.Then du = d/dx(sin x) dx = cos x dx.Substitute u and du into the integral:= ∫ (1 - u2) duNow, integrate with respect to u:= u - (u3 / 3) + CFinally, substitute u = sin x back into the expression:= sin x - (sin3 x / 3) + CThis matches option A.

Hint

Use the identity cos2x = 1 - sin2x and then a u-substitution where u = sin x.

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 742 of 930
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